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Energy

Energy. Conservative system E = T + V ( q ) Solve for the velocity. Position can be found analytically for some V ( q ). General solution is numeric. Oscillator in One Coordinate. Potential Energy. Turning points q 1 , q 2 v q = 0 V ( q 1 ) = V( q 2 ) = E

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Energy

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  1. Energy

  2. Conservative system E = T+ V(q) Solve for the velocity. Position can be found analytically for some V(q). General solution is numeric. Oscillator in One Coordinate

  3. Potential Energy • Turning points q1, q2 • vq = 0 • V(q1) = V(q2) = E • T > 0, so V(q1 < q < q2) < E • Equilibrium qmin • The period depends on E only when potential is not a parabola. V(q) E q q1 qmin q2

  4. Single variable is the angle s = lq q = q V(q) = mgl(1 – cosq) Small oscillation limit F = –(mg/l) s Harmonic motion Finite amplitude Make substitutions Consider E < 2, E = 2, E > 2 Plane Pendulum

  5. Bound Motion • Energy below threshold • E < 2 • Turning points exist • Solution is an elliptic integral • Approximate period

  6. Critical Energy • Energy at threshold • E = 2 • Non-periodic motion • Non-circular motion • Reaches peak at infinite time

  7. Unbound Motion • Energy above threshold • E > 2 • Non-uniform circular motion • Solution is an elliptic integral • Period depends on energy • and acceleration of gravity

  8. Phase Portrait • A plot of position vs. velocity. • Phase space is something more detailed. E > 2 E = 2 E < 2

  9. Damped Oscillator • Small damping factor l • Depends on velocity • Total energy is decreasing • Find q, q’ by usual means • Compare periods • One cycle T • Energy loss in that time next

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